Graphs with Constant Sum of Domination and Inverse Domination Numbers

T Tamizh Chelvam,T Asir

doi:10.1155/2012/831489

T Tamizh Chelvam, T Asir

Open Access

https://doi.org/10.1155/2012/831489

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Abstract

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

Highlights

The inverse domination number γ G is the minimum cardinality of the inverse dominating sets.Domke et al 2004 characterized connected graphs G with γGγGn, where n is the number of vertices in G
For D ⊆ V, if every vertex in V − D is adjacent to at least one vertex in D, D is said to be a dominating set of G 1
The minimum cardinality among all dominating sets of G is called domination number of G, and it is denoted by γ G

Summary

Introduction

The inverse domination number γ G is the minimum cardinality of the inverse dominating sets.Domke et al 2004 characterized connected graphs G with γGγGn, where n is the number of vertices in G. Any graph with no isolated vertices contains an inverse dominating set.

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